if 5tan⁡ϕ=tan⁡θ then the maximum value of 5tan2⁡(θ−ϕ) is

# if $5\mathrm{tan}\varphi =\mathrm{tan}\theta$ then the maximum value of $5{\mathrm{tan}}^{2}\left(\theta -\varphi \right)$ is

1. A

1

2. B

4

3. C

6

4. D

9

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)

### Solution:

$t=\mathrm{tan}\varphi ,\mathrm{tan}\left(\theta -\varphi \right)=\frac{5t-t}{1+5{t}^{2}}=\frac{4t}{1+5{t}^{2}}$

5${\mathrm{tan}}^{2}\left(\theta -\varphi \right)=\frac{80{t}^{2}}{{\left(1+5{t}^{2}\right)}^{2}}=\frac{80}{{\left(\frac{1}{t}+5t\right)}^{2}}\le \frac{80}{\left(2\sqrt{5}{\right)}^{2}}=4$

since$A.M$ $\ge$$G.M.$

Register to Get Free Mock Test and Study Material

+91

Verify OTP Code (required)